Research Article

Advances in Mechanical Engineering2017, Vol. 9(9) 1–10� The Author(s) 2017DOI: 10.1177/1687814017720053journals.sagepub.com/home/ade

Multi-objective optimization design ofcycloid pin gear planetary reducer

Yaliang Wang, Qijing Qian, Guoda Chen, Shousong Jinand Yong Chen

AbstractA multi-objective optimal model of a K-H-V cycloid pin gear planetary reducer is presented in this article. The optimalmodel is established by taking the objective functions of the reducer volume, the force of the turning arm bearing, andthe maximum bending stress of the pin. The optimization aims to decrease these objectives and obtains a set of Paretooptimal solutions. In order to improve the spread of the Pareto front, the density estimation metric (crowding distance)of non-dominated sorting genetic algorithm II is replaced by the k nearest neighbor distance. Then, the improved algo-rithm is used to solve this optimal model. The results indicate that the modified algorithm can obtain the better Paretooptimal solutions than the solution by the routine design.

KeywordsCycloid speed reducer, planetary transmission, multi-objective optimization, evolutionary algorithm, density estimation

Date received: 11 January 2017; accepted: 15 June 2017

Academic Editor: Ismet Baran

Introduction

Cycloid speed reducers have some excellent characteris-tics, such as compact structure, wide scope of ratios,high transmission efficiency, low noise, and smooth-ness. Thus, they are widely used in all kinds of mechan-ical transmission. However, many design parametersand complex constraints in the reducer design bringmore difficulty. At present, a lot of researches are avail-able in this area. Chen et al.1 established the equationof meshing for small teeth difference planetary gearingand a universal equation of cycloid gear tooth profilebased on cylindrical pin tooth and given motion. Heet al.2 carried out optimum design and experiment onthe double crank ring-plate-type pin-cycloid planetarydrive to reduce its noise and vibration. Sensinger3 pre-sented a unified design method to optimize cycloidaldrive profile, efficiency, and stress. Blagojevic et al.4

introduced a new two-stage cycloid speed reducer,which was characterized by good load distribution anddynamic balance. Blagojevic et al.5 found the friction

between cycloid disk and housing rollers affected thecontact force, friction torque, and transmission effi-ciency. Hsieh6 proved the nonpinwheel design ofcycloid speed reducer could effectively reduce vibra-tion, stress value, and stress fluctuation.

In the engineering design, many multi-objective opti-mization problems (MOOPs) can be found. Some or allobjectives often conflict with each other; in otherwords, all objectives cannot achieve its own best valuesimultaneously.7 Thus, the policymakers need to choosethe compromise design parameters according to the

Key Laboratory of Special Purpose Equipment and Advanced Processing

Technology, Ministry of Education, Zhejiang University of Technology,

Hangzhou, China

Corresponding author:

Shousong Jin, Key Laboratory of Special Purpose Equipment and

Advanced Processing Technology, Ministry of Education, Zhejiang

University of Technology, 18 Chaowang Road, Hangzhou 310014, China.

Email: jinshs@zjut.edu.cn

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License

(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/

open-access-at-sage).

reality. For the last decade and more, many classicalmulti-objective evolutionary algorithms (MOEAs) werestudied, such as the non-dominated sorting geneticalgorithm II (NSGA-II),8 the improved strength Paretoevolutionary algorithm (SPEA2),9 and the multi-objective particle swarm optimization (MOPSO).10 Asan efficient algorithm, NSGA-II has been widely usedto solve MOOPs.11–13 Some researchers used NSGA-IIto optimize gear reducers. Deb and Jain14 provedMOEAs can solve multi-speed gearbox design optimi-zation problem with more than one optimal goal anddifferent kinds of design parameters. Tripathi andChauhan15 applied NSGA-II to optimize the volume ofgearbox and the surface fatigue life factor simultane-ously. Sanghvi et al.16 used NSGA-II to solve optimiza-tion design problem of a two-stage helical gear train, inwhich the bearing force and volume were optimized.Three optimization methods (MATLAB optimumtools, genetic algorithm, and NSGA-II) were com-pared. It was shown that NSGA-II obtained betteroptimization objective values than other methods. Liet al.17 combined NSGA-II and fuzzy set theory tooptimize dynamic model of steering mechanism and gotthe better result than the original design. However, fartoo little attention has been paid to the research aboutmulti-objective optimal design of cycloid speed redu-cers. Yu et al.18 and Yu and Xu19 selected volume asthe optimization objective. Xi et al.20 selected volumeand efficiency as optimization objectives, and the multi-objective optimization model was solved by transform-ing it into a single-objective problem, which ignored therelationship between objectives. Wang et al.21 focusedon reducing the volume and improving the efficiency ofreducers, and the optimal model was solved by theMATLAB genetic algorithm toolbox.

In this article, a multi-objective optimization modelof reducer is established to minimize the volume of thereducer, force of the turning arm bearing, and maximalbending stress of the pin. It is expected to provide a newway for multi-objective optimization design of cycloidreducer and similar structural optimization problems.Moreover, in order to improve the spread of the Paretofront, the density estimation metric of NSGA-II isreplaced by the k nearest neighbor distance.

Optimization design model of the K-H-Vcycloid driver

Figure 1 shows the typical structure of the cycloidalpinwheel transmission, which mainly consists of the fol-lowing parts:

1. Planet carrier. It is composed of two parts,namely, the input shaft and the dual eccentricsleeves.

2. Pin gear (also called pinwheel). It is uniformlydistributed over the circumferential direction.The pin gear consists of the pin and pin sleeve.

3. Cycloid gear. To keep the static balance of theinput shaft and increase the loading capacity ofthe reducer, two identical cycloid gear structuresare usually adopted. The cycloid gears areinstalled on the dual eccentric sleeves, and theirposition differs 180�. In order to reduce the fric-tion between eccentric sleeve and cycloid gear,the turning arm bearing is installed between thetwo parts.

4. Output mechanism. This reducer often uses theoutput mechanism that called pin axle type.There are some cylindrical pin holes on thecycloid gear for inserting the cylindrical pins(the cylindrical pin sleeve is mounted on thecylindrical pin). Thus, the rotation motion ofcycloid gears can be output by cylindrical pins.

Theoretical tooth profile of cycloid gear

As shown in Figure 2, rolling circle 2 is the circum-scribed circle of base circle 1, and the center of base cir-cle 1 (Og) is also the origin of rectangular coordinatesystem. According to the forming principle of cycloidgear profile, supposing the base circle 1 is fixed, thelocus of point M in circle 2 is a epicycloid when circle 2scrolls from tangent point A to tangent point B. Therotation angle of the circle 2 (with the center O) aroundthe circle 1 (with the center Og) is denoted by u. Therotation angle of circle 2 is denoted by ub and the abso-lute angle of circle 2 is denoted by uh. The coordinatesof a point on the theoretic profile of cycloidal gear canbe expressed as

x0 =Rz sinu� esinuh

y0 =Rz cosu� ecosuh

�ð1Þ

Figure 1. Typical structure of cycloidal pinwheel transmission.

2 Advances in Mechanical Engineering

where Rz is the radius of the pin gear distributed circle,e is the eccentric distance (e = ObOg = (K1Rz)=Zb), K1

is the short amplitude coefficient (K1 =(rb=Rz), rb is thepitch radius of the pinwheel), and Zb is the tooth num-ber of the pin gears (Zb = (uh=u)). Therefore, equation(1) can be expressed as

x0 =Rz sinu� K1

ZbsinZbu

� �y0 =Rz cosu� K1

ZbcosZbu

� �9=; ð2Þ

According to the radius of curvature formula, theradius of theoretical tooth profile of cycloid gear canbe expressed as

r0 =(1+K2

1 � 2K1cosub)32RZ

K1(1+Zb) cos ub � (1+ ZbK21 )

ð3Þ

Meshing force of pin gear and cycloid gear

As shown in Figure 3, the angular velocity (vg) ofcycloid gear has the opposite direction on output tor-que when pin gears are fixed. In the switching mechan-ism, the rotation direction of cycloid gear and pinwheelare same. On the left side of the Y axis, pinwheels andcycloid gear mesh, and the direction of force (Fi) of thepin gear on the cycloid gear intersect with node P. Onthe right side of the Y axis, pin gears leave cycloid gear,and there is no force between them. Supposing Tg rep-resents the resistance torque of each cycloid gear, whena torque with the equal value and opposite direction ofTg is applied on the cycloid gear, the center of pin gearhas a tiny circumferential displacement Du. The size of

Fi is proportional to component of Du in the directionof Fi. That is

Fi}Ducosai =Duli

RZð4Þ

where li is the vertical distance between Ob and thedirection of Fi. When li is equal to rb, and Fi is equal toFmax. That is

Fmax } Durb

Rzð5Þ

In the triangle ObBP, sin ui =(li=rb). According toequations (4) and (5), Fi can be expressed as

Fi =Fmaxli

rb=Fmax sin ui ð6Þ

The maximum engaging force22 is shown as

Fmax=4Tg

K1ZgRzð7Þ

where Zg is the tooth number of the cycloid gear. Ingeneral, Tg = 0:55Tv, and Tv is the output torque.Therefore, Fmax can be expressed as

Fmax=2:2Tv

K1ZgRzð8Þ

In the triangle PObOi, according to sine theorem,sin ui can be expressed as

sin ui =Rz

POi

sin ubi ð9Þ

Figure 2. Theoretical tooth profile of cycloid gear. Figure 3. Meshing force of pin gear and cycloid gear.

Wang et al. 3

Besides, according to cosine theorem, POi can beexpressed as

POi =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2z + r2

b � 2Rzrb cos ubi

qð10Þ

Because K1 is equal to (rb=Rz), POi can be expressedas

POi =Rz

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+K2

1 � 2K1 cos ubi

qð11Þ

According to equations (6), (8), (9), and (11), Fi canbe expressed as

Fi =2:2Tv sin ubi

K1ZgRz

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+K2

1 � 2K1 cos ubi

q ð12Þ

Design variables

A vector which consists of five design variables isexpressed as X =(Dz, d

0z,B,K1, d

0w), where Dz is the dia-

meter of the pin gear distributed circle, d0z is the dia-

meter of the pin, B is the width of the cycloid gear, andd0w is the diameter of the cylindrical pin.

Objective functions

For the convenience, the input power, input speed,transmission ratio, turning arm bearing, and the num-ber of cylindrical pins are given. Under these premises,the sub-objectives are presented as follows.

Volume of reducer. The radial dimension is affected bythe diameter of the pin gear distributed circle (Dz) andthe diameter of the pin sleeve. The axle dimension isaffected by the width of the cycloid gear (B) and thegap between two cycloid gears. Therefore, the volumecan be expressed as20

min f1(X )=p

4(Dz + d

0

z + 2D1)2(2B+ d) ð13Þ

where D1 is the thickness of pin sleeve and d is the gapbetween the two cycloid gears. In general, d= b� B (bis the width of the turning arm bearing).

Radial load on turning arm bearing. The service life of turn-ing arm bearing largely depends on its radial load. Itfurther affects the service life of the speed reducer.Hence, minimizing the radial load of the turning armbearing should be considered. It can be expressed as

min f2(X )=2:6TgZb

K1DzZgð14Þ

Maximal bending stress of the pin. The break in pin is oneof the main failure forms of this cycloid reducer. Hence,minimizing the maximal bending stress of the pin is thethird sub-objection. For pin gear which has two ful-crums (in Figure 4), the bending stress of the pin isshown as follows

sF ’Mwmax

0:1d0z

3=

44L1L2Tv

LK1ZgDzd0z

3ð15Þ

where L1 = 0:5B+ d0+0:5D, L2 = 1:5B+d

0+d+0:5D,

and L= L1 + L2 = 2B+ 2d0+ d+D. D is the thick-

ness of the side wall of a pin gear housing. d0is the

interval between cycloid gear and the internal face ofthe side wall, and in general, d

0z�D�B. The third sub-

objective is presented as

min f3(X )=44L1L2Tv

LK1ZgDzd0z

3ð16Þ

Constraint conditions

Short amplitude coefficient. If the short amplitude coeffi-cient is bigger, the minimal radius of theoretical toothprofile of cycloid gear will be reduced, which willdecrease the outer radius of the pin sleeve, that is, thecontact strength of the cycloid gear and the pin gearincreases. According to equation (16), the bendingstress of the pin will increase with the decrease in theshort amplitude coefficient. It suggests that the con-straint range of the short amplitude coefficient is [0.45,0.8];20 thus, the constraint equation can be defined by

g1(X )= 0:45� K1� 0 ð17Þ

g2(X )=K1 � 0:8� 0 ð18Þ

Cycloid tooth profile. To prevent cycloid tooth profilefrom undercut and sharp angle, the ratio of an externaldiameter of the pin sleeve to a diameter of the pin gear

Figure 4. Typical structure of a pin gear with two pivots.

4 Advances in Mechanical Engineering

distributed circle should be less than the minimum coef-ficient of the theoretical tooth profile curvature radius(amin). Accordingly, this constraint can be defined by

g3(X )=(d0z + 2D1)

Dz� amin� 0 ð19Þ

where amin = (1+K1)2=(1+K1 + ZgK1).

Maximum diameter of the cylindrical pin hole. In order toguarantee the strength of the cycloid gear, there mustbe a certain thickness (T ) between two adjacent cylind-rical pin holes, and in general, T = 0:03Dz. Thus, themaximum diameter of the cylindrical pin hole meets thefollowing constraints

g4(X )= 2T � Dw + dsk +D1� 0 ð20Þ

g5(X )= T � Dw sinp

Zw+ dsk� 0 ð21Þ

where Zw is the number of cylindrical pins. Dw is thediameter of the cylindrical pin hole distributed circle,which can be defined by

Dw =dfc +D1

2ð22Þ

where dfc is the diameter of the root circle of a cycloidgear. D1 is the diameter of the cycloid gear center hole,which is also the external diameter of the turning armbearing.

dsk is the diameter of the cylindrical pin hole, whichcan be defined by

dsk = dw + 2e ð23Þ

where dw is the external diameter of a cylindrical pinsleeve.

Pin-diameter coefficient. In order to guarantee thestrength of the pin gear housing and avoid the interfaceof pin gears, the value of pin-diameter coefficient (K2)

22

should be in the range of 1.25–4. Thus, this constraintcondition can be expressed as

g6(X )= 1:25� K2� 0 ð24Þ

g7(X )=K2 � 4� 0 ð25Þ

where K2 =(Dz=(d0z + 2D1)) sin (p=Zb).

Contact strength of the cycloid gear and the pin gear. To pre-vent the tooth surface from scuffing failure and fatiguepitting, the meshing between the pin teeth and thecycloidal gear teeth should meet the contact strength.Using Hertz theory, the contact stress (sH) between thepin gear and the cycloid gear can be expressed as

sH = 0:418

ffiffiffiffiffiffiffiffiffiffiFiEd

Brd

sð26Þ

where Fi is the meshing force in a certain positionbetween the pin gear and the cycloid gear, rd is theequivalent curvature radius of the contact point, andEd is the equivalent elastic modulus between the pingear and the cycloid gear. Since both materials areGCr15 bearing steel, Ed is equal to 2:10 3 105 MPa.The constraint can be defined by

g8(X )= 0:418

ffiffiffiffiffiffiffiffiffiffiFiEd

Brd

s� sHP� 0 ð27Þ

where sHP is the allowable contact stress.

Bending strength of the pin gear. According to equation(15), the constraint is as follows

g9(X )=44L1L2Tv

LK1ZgDzd0z

3� sFP� 0 ð28Þ

where sFP is the allowable bending stress.

Contact strength between the cylindrical pin and the cylindricalpin hole. According to Rao,22 this constraint is asfollows

g10(X )= 0:0949

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10K1TvDz

ZwDwB(r2wZb +

DzK1

2rw)

s� sHP� 0

ð29Þ

where rw is the radius of the cylindrical pin sleeve,which can be defined by

rw =d0w

2+D2 ð30Þ

where D2 is the thickness of the cylindrical pin sleeve.

Bending strength of the cylindrical pin. According to Rao,22

this constraint can be expressed as

g11(X )=96Tv(1:5B+ d)

ZwRwd0w

3� sFP� 0 ð31Þ

Life of the turning arm bearing

g14(X )= Lh �106

60n

C

F

� �103

� 0 ð32Þ

where Lh is the turning arm bearing life, and Lh is usuallyequal to 5000h; n is the rotating speed of the bearing; C

is the dynamic load rating; F is the equivalent dynamicalload; and F = 1:2R (R is the radial load on bearing).

Wang et al. 5

Improved NSGA-II (NSGAN)

Non-dominated ranking and crowding distance areadopted in NSGA-II to control the evolutionary popu-lation size. The density estimation technique of thecrowding distance calculates the average distance oftwo points on either side of this point along each of theobjectives. Sometimes, this method cannot completelyreflect the crowdedness between individuals. As shownin Figure 5, supposing that d represents the unit of adistance in objective space, the distance of solution A(12d) is equal to that of the solution B (12d) accordingto crowding distance. And, it can also be seen that B ismore crowded than A. However, the density estimationmetric of the k nearest neighbor distance9 can avoidthis case. Through the k nearest neighbor distance, thefirst nearest neighbor distance of B (2

ffiffiffi2p

d) is smallerthan that of A (3

ffiffiffi2p

d), which means B is more crowdedthan A. Hence, the density estimation metric of the knearest neighbor distance is introduced to improve thespread of Pareto front, and the improved algorithm ismarked as NSGAN. The k nearest neighbor distancecomputation procedure is shown in Table 1.

Optimization design example

GCr15 material is selected for pin, pin sleeve, cycloidgear, cylindrical pin, and cylindrical pin sleeve. Themain model parameters22 are shown in Table 2.

NSGA-II and NSGAN are adopted to optimize thismodel, respectively. The parameters in algorithms arelisted as following: population size N is 100, evolu-tional generation is 600, crossover probability pc is 0.9,and mutation probability pm is 1/n, where ‘‘n’’

represents the number of the design variables. The twoalgorithms run 30 times independently. According toTable 2 and Rao,22 the range of design variables isshown in Table 3.

The optimization procedure is shown in Figure 6. Atfirst, an initial population is randomly created in therange of design variables, and the solutions utilize real-number encoding. Second, compute the rank and knearest neighbor distance for the every individual inpopulation. Then, the binary tournament selection,recombination, and mutation operators are used to cre-ate an offspring population. In a binary tournamentselection, a lower rank and bigger k nearest neighbordistance is the selection criteria. NSGAN uses simu-lated binary crossover and polynomial mutation. Afterthat, the next population is selected from the offspringand previously population based on the rank and k

Figure 5. Pareto front.

Table 1. Pseudocode of k nearest neighbor distance.

k nearest neighbor distance

% chromosome consists of the decision variables, value of the objective functions and rank, and the rank of every individual is basedon non-domination.% V is the dimension of decision variable space.% M is the dimension of the objective space.% population consists of the decision variables, value of the objective functions, rank and k nearest neighbor distance.

obj = chromosome (:,V+ 1:M+V);for i = 1: M

obj(:,i) = obj(:,i)/(max(obj(:,i))-min(obj(:,i))); %normalize sub-objectivesend forn= chromosomej j; %number of solutions (individuals) in chromosomefor j = 1: n–1

for m = j+ 1:nEj,m = obj(j, : )� obj(m, : )k k2; %the Euclidean distance between two individuals in objective space.E(m,j) = E(j,m);

end forE(j,j) = 0; %E stores the k nearest neighbor distance of every individual

end forSE = sort(E,2); %sort the distance in ascending orderpopulation = zeros(n, M+V+ n);population (:,1: M+V+ 1) = chromosome;population (:,M+V+ 2:M+V+ n) = SE(:,2:n);

6 Advances in Mechanical Engineering

nearest neighbor distance. If the maximal generation isreached, output the result; if not, keep on evolving.

Spread test

There are three sub-objectives, and the spread indica-tor23 is chosen to measure the spread of the Pareto frontcorresponding to obtained solutions. The smaller thisvalue is, the more uniform the solutions distribute. Theindicator is defined as

D=

Pmi= 1

d(Ei,O)+P

X2O d(X ,O)� d�� ��

Pmi= 1

d(Ei,O)+ ( Oj j � m)d

ð33Þ

where O is a set of solutions, (E1, . . . ,Em) are m extremesolutions in the set of Pareto optimal solutions, m is thenumber of objectives and

d(X ,O)= minY2O, Y 6¼X

F(X )� F(Y )k k ð34Þ

d =1

Oj jX

X2O d(X ,O) ð35Þ

Three extreme solutions (i.e. these solutions have amaximal sub-objective value) are selected among theobtained solutions based on the two algorithms. Theirobjective values are shown as follows: E1 (3.11e–3,4746.33, 34.44), E2 (1.49e–3, 10756.94, 94.89), and E3(1.43e–3, 8444.82, 150.00). Among them, E1 and E2come from NSGAN, and E3 comes from NSGA-II.

Compute the spread indicators of the Pareto frontobtained by two algorithms. The Pareto fronts obtainedby the two algorithms are shown in Figures 7 and 8.Spread indicator statistical result is shown in Table 4.

From Figures 7 and 8, it can be shown that theNSGAN gets a more uniform Pareto front than thatby NSGA-II. Table 3 shows a statistical result of thespread indicator. These data demonstrate that NSGANobtains better result than NSGA-II in terms of thespread. In order to prove NSGAN has a significantrole in the spread, a t-test is performed for the averageof the spread indicator by software SPSS. The signifi-cance level is supposed as 0.05. The t statistic is 53.513,and the corresponding two-tailed probability is 0.000.

Table 2. Model parameters.

Parameters Values

Input power (kw) 4Input speed (r/min) 1440Transmission ratio 29Number of the cylindrical pins, Zw 10Contact stress, sHP (MPa) 850Bending stress, sFP (MPa) 150External diameter of turning arm bearing, D1 (mm) 86.5Width of turning arm bearing, B (mm) 25Rated dynamic load of the turning arm bearing, C (N) 64,900

Table 3. Range of design variables.

Design variables Interval Unit

Diameter of pin gear distributed circle 200<Dz<300 mmDiameter of pin 8<d

0

z<12 mmWidth of cycloid gear 12<B<17 mmShort width coefficient 0:45<K1<0:8 –Cylindrical pin 12<d

0w<55 mm

Figure 6. Flowchart of optimization procedure based onNSGAN.

Figure 7. Pareto front obtained by NSGA-II.

Wang et al. 7

Because the two-tailed probability is smaller than thesignificance level, the null hypothesis is refused. Itshows that two algorithms have significant differencesin spread indicator.

Results analysis

A total of 100 non-dominated solutions are picked outfrom 3000 solutions gained by NSGAN according torank and k nearest neighbor distance. The minimal sub-objective solutions (volume, radial load of the turningarm bearing, or maximal bending strength of the pin)are selected out from 100 non-dominated solutions.These solutions are Solution 1 (1.37e–3, 9188.71,149.93), Solution 2 (2.92e–3, 4741.87, 40.96), andSolution3 (2.82e–3, 4853.17, 33.50). From three extremesolutions, three sub-objectives cannot reach minimumtogether. Figure 8 presents the Pareto front of 100 non-dominated solutions. In addition, the sub-objectives ofthe routine design22 are also shown in Figure 9.

Next, the solutions for which the three sub-objectivesare all smaller than that of the routine design are pickedout from Figure 9, which are shown in Figure 10. Thereare eight total solutions whose three sub-objectives aresmaller than that of the routine design. Table 5 presentsthe design variables and sub-objectives of these eightsolutions and the routine design. Besides, the designvariables are rounded under the premise of meeting theconstraints.

Through the above analysis, NSGAN can obtainsolutions for which the three sub-objectives are super-ior to that of the routine design. However, in the prac-tical design, due to some sub-objectives conflict witheach other, the designer has to make a compromise.For example, the designers can minimize the volume ofthe reducer and maximal bending stress of the pin, butappropriately increase the radial load of the turningarm bearing under the constraints, i.e., make a compro-mise. Depending on this situation, some preferencesolutions are selected from the 100 non-dominatedsolutions. For the better comparison, Figure 11 showsthe relationship between volume and radial load of theturning arm bearing, and Figure 12 illustrates the rela-tionship between volume and maximal bending stressof the pin. When the volume increases, the radial loadof the turning arm bearing will be reduced. Moreover,according to equations (14) and (15), the f1 (volume)and f2 (radial load of the turning arm bearing) are

Figure 8. Pareto front obtained by NSGAN.

Table 4. Average and standard deviation of the spreadindicator.

Algorithm Average Standard deviation

NSGA-II 0.6302 4.82e–2NSGAN 0.1431 1.28e–2

Figure 9. Non-dominated Pareto front obtained by NSGAN.

Figure 10. Three sub-objectives are smaller than that of theroutine design.

8 Advances in Mechanical Engineering

conflicting. The increase of volume can reduce the max-imum bending stress of the pin to a certain extent.However, the maximum bending stress of the pin is notentirely dependent on volume, and equation (16) showsthat short amplitude coefficient can also have impacton maximal bending stress of the pin.

In Figures 11 and 12, there are five magenta prefer-ence solutions. As shown in Figure 11, the second sub-objective (radial load on bearing) is bigger than that ofthe routine design, but the first sub-objective (volume)is smaller than that of the routine design. Meanwhile,the third sub-objective (maximal bending strength ofthe pin) is also smaller than that of the routine design

according to Figure 12. These five preference solutionssacrifice the second sub-objective, but the first sub-objective is obviously improved.

Conclusion

A multi-objective optimization model of a cycloid pingear planetary reducer with the goals of minimizing thevolume, the radial load on turning arm bearing, and themaximal bending stress of the pin is considered in thisarticle. The density estimation technique of NSGA-II isimproved using the k nearest neighbor distance. The

Table 5. Design variables and sub-objectives.

Design method Design variables Sub-objectives

Diameter ofpin geardistributed circle,Dz (mm)

Diameter ofpin, d

0z (mm)

Width ofcycloid gear,B (mm)

Short widthcoefficient, K1

Cylindricalpin, d

0w (mm)

f1 (m3) f2 (N) f3 (MPa)

Routine design 240 10 17 0.75 20 2.16e–3 6322 80.38NSGANRounded values

233.313233

9.5269.5

12.00012.0

0.7980.80

21.72621.7

1.80e–31.80e–3

61126105

78.0878.57

NSGANRounded values

242.837243

9.4699.5

12.00012.0

0.8000.80

22.29722.3

1.94e–31.94e–3

58585854

76.0775.33

NSGANRounded values

232.986234

10.50110.5

12.35012.4

0.7870.79

19.91620.0

1.83e–31.83e–3

62076156

61.3860.98

NSGANRounded values

251.541252

9.4479.4

12.00012.0

0.8000.80

20.54520.5

2.07e–32.08e–3

56555645

73.9074.78

NSGANRounded values

244.282244

9.9139.9

12.00012.0

0.8000.80

19.07919.0

1.97e–31.96e–3

58235830

66.7367.04

NSGANRounded values

239.075239

11.33211.3

12.93213.0

0.7990.80

21.98822.0

1.96e–31.96e–3

59585952

48.6649.07

NSGANRounded values

252.159252

10.35510.4

12.00012.0

0.7990.80

20.48220.5

2.10e–32.09e–3

56495645

57.4856.77

NSGANRounded values

251.534252

11.17011.2

12.00212.0

0.7980.80

22.28322.3

2.10e–32.11e–3

56705645

46.9946.44

Figure 11. Volume of the reducer and radial load of theturning arm bearing.

Figure 12. Volume of the reducer and maximal bendingstrength of the pin.

Wang et al. 9

improved algorithm (NSGAN) is used to solve themulti-objective optimization design model. Althoughsome sub-objectives conflict with each other, theMOEA can optimize these sub-objectives simultane-ously. NSGAN obtains more uniform Pareto frontsthan NSGA-II, and the Pareto front can present therelationship between sub-objectives, which is more con-venient for the designers to select design solutions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This work was supported, in part, by a grant fromNatural Science Foundation of Zhejiang Province of China(Nos LY16G010013, LY17E050023, and LQ16E050012),National High-Tech R&D Program of China (No.2015AA043002), and National Natural Science Foundationof China (No. 71371170).

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